# Theory of the electron-ion temperature relaxation rate spanning the hot solid metals and plasma phases

## TL;DR

This paper develops a comprehensive quantum-mechanical theory for electron-ion energy exchange rates applicable across hot solids and plasmas, unifying and extending existing models with broad applicability.

## Contribution

It derives a general expression for the electron-ion coupling factor that includes quantum, thermal, disorder, and correlation effects, unifying various limiting models.

## Key findings

- Reduces to the standard electron-phonon coupling in hot solids.
- Extends the electron-phonon model beyond harmonic approximation.
- Recovers the Spitzer formula and Fermi golden rule in plasma regimes.

## Abstract

We present a theory for the rate of energy exchange between electrons and ions -- also known as the electron-ion coupling factor -- in physical systems ranging from hot solid metals to plasmas, including liquid metals and warm dense matter. The paper provides the theoretical foundations of a recent work [J. Simoni and J. Daligault, Phys. Rev. Lett. {\bf 122}, 205001 (2019)], where first-principles quantum molecular dynamics calculations based on this theory were presented for representative materials and conditions. We first derive a general expression for the electron-ion coupling factor that includes self-consistently the quantum mechanical and statistical nature of electrons, the thermal and disorder effects, and the correlations between particles. We show that our theory reduces to well-known models in limiting cases. In particular, we show that it simplifies to the standard electron-phonon coupling formula in the limit of hot solids with lattice and electronic temperatures much greater than the Debye temperature, and that it extends the electron-phonon coupling formula beyond the harmonic phonon approximation. For plasmas, we show that the theory readily reduces to well-know Spitzer formula in the hot plasma limit, to the Fermi golden rule formula in the limit of weak electron-ion interactions, and to other models proposed to go beyond the latter approximation. We explain that the electron-ion coupling is particularly well adapted to averaged atom models, which offer an effective way to include non-ideal interaction effects to the standard models and at a much reduced computational cost in comparison to first-principles quantum molecular dynamics simulations.